Let $\Sigma _g$ be a closed oriented surface of genus g. In this article, we discuss how to define coloring invariants and their generalizations for links in $\Sigma _g\times S^1$.

We explore the relationship between (3-isogeny induced) Selmer group of an elliptic curve and the (3 part of) the ideal class group, over certain non-abelian number fields.

Let $2\leq p<\infty $ and X be a complex infinite-dimensional Banach space. It is proved that if X is p-uniformly PL-convex, then there is no nontrivial bounded Volterra operator from the weak Hardy space $\mathscr {H}^{\text {weak}}_p(X)$ to the Hardy space $\mathscr {H}^+_p(X)$ of vector-valued Dirichlet series. To obtain this, a Littlewood–Paley inequality for Dirichlet series is established.

The blow-up of the anticanonical base point on a del Pezzo surface S of degree 1 gives rise to a rational elliptic surface $\mathscr {E}$ with only irreducible fibers. The sections of minimal height of $\mathscr {E}$ are in correspondence with the $240$ exceptional curves on S. A natural question arises when studying the configuration of these curves: if a point on S is contained in “many” exceptional curves, is it torsion on its fiber on $\mathscr {E}$? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree $2$, where there are 56 exceptional curves, that if “many” equals $4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if ‘many’ equals $9$ or more. Moreover, we give counterexamples where a non-torsion point lies in the intersection of $7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves.

We give sufficient conditions for the essential spectrum of the Hermitian square of a class of Hankel operators on the Bergman space of the polydisc to contain intervals. We also compute the spectrum in case the symbol is a monomial.

We give necessary and sufficient conditions for certain pushouts of topological spaces in the category of Čech’s closure spaces to agree with their pushout in the category of topological spaces. We prove that in these two categories, the constructions of cell complexes by a finite sequence of closed cell attachments, which attach arbitrarily many cells at a time, agree. Likewise, the constructions of CW complexes relative to a compactly generated weak Hausdorff space that attach only finitely many cells, also agree. On the other hand, we give examples showing that the constructions of finite-dimensional CW complexes, CW complexes of finite type, and relative CW complexes that attach only finitely many cells, need not agree.

We show that for $5/6$-th of all primes p, Hilbert’s 10th problem is unsolvable for the ring of integers of $\mathbb {Q}(\zeta _3, \sqrt [3]{p})$. We also show that there is an infinite set S of square-free integers such that Hilbert’s 10th problem is unsolvable over the ring of integers of $\mathbb {Q}(\zeta _3, \sqrt {D}, \sqrt [3]{p})$ for every $D \in S$ and for every prime $p \equiv 2, 5\ \pmod 9$. We use the CM elliptic curves $y^2=x^3-432 D^2$ associated with the cube-sum problem, with D varying in suitable congruence class, in our proof.

A convex body R in the hyperbolic plane is called reduced if any convex body $K\subset R$ has a smaller minimal width than R. We answer a few of Lassak’s questions about ordinary reduced polygons regarding its perimeter, diameter, and circumradius, and we also obtain a hyperbolic extension of a result of Fabińska.

This article describes Hilbert spaces contractively contained in certain reproducing kernel Hilbert spaces of analytic functions on the open unit disc which are nearly invariant under division by an inner function. We extend Hitt’s theorem on nearly invariant subspaces of the backward shift operator on $H^2(\mathbb {D})$ as well as its many generalizations to the setting of de Branges spaces.

I examine some analytical properties of a nonlinear reaction-diffusion system that has been used to model the propagation of a wildfire. I establish global-in-time existence and uniqueness of bounded mild solutions to the Cauchy problem for this system given bounded initial data. In particular, this shows that the model does not allow for thermal blow-up. If the initial temperature and fuel density also satisfy certain integrability conditions, the spatial $L^2$-norms of these global solutions are uniformly bounded in time. Additionally, I use a bootstrap argument to show that small initial temperatures give rise to solutions that decay to zero as time goes to infinity, proving the existence of initial states that do not develop into traveling combustion waves.

In this paper, we show existence of bountiful examples of Gorenstein local rings A and B such that there is a triangle equivalence between the stable categories CM(A), CM(B).

We compute the large size limit of the moment formula derived in [14] for the Hermitian Jacobi process at fixed time. Our computations rely on the polynomial division algorithm which allows to obtain cancellations similar to those obtained in [3, Lemma 3]. In particular, we identify the terms contributing to the limit and show they satisfy a double recurrence relation. We also determine explicitly some of them and revisit a special case relying on Carlitz summation identity for terminating $1$-balanced ${}_4F_3$ functions taken at unity.

A coset partition of a group G is a set partition of G into finitely many left cosets of one or more subgroups. A driving force in this research area is the Herzog–Schönheim Conjecture [15], which states that any nontrivial coset partition of a group contains at least two cosets with the same index. Although many families of groups have been shown to satisfy the conjecture, it remains open.

A Steiner coset partition of G, with respect to distinct subgroups $H_1,\dots ,H_r$, is a coset partition of G that contains exactly one coset of each $H_i$. In the quest of a more structural version of the Herzog–Schönheim Conjecture, it was shown that there is no Steiner coset partition of G with respect to any $r\geq 2$ subgroups $H_i$ that mutually commute [1]. In this article, we show that this result holds for $r=4$ mutually commuting subgroups provided that G does not have $C_2\times C_2\times C_2$ as a quotient, where $C_2$ is the cyclic group of order $2$. We further give an explicit construction of Steiner coset partitions of the n-fold direct product $G^*=C_p\times \ldots \times C_p$ for p prime and $n\geq 3$. This construction lifts to every group extension of $G^*$.

We characterize the compact elements and the hypocompact radical of a crossed product $C_0(X)\times _\phi \mathbb Z$, where X is a locally compact metrizable space and $\phi :X\rightarrow X$ is a homeomorphism, in terms of the corresponding dynamical system $(X,\phi )$.

Isoclinic subspaces have been studied for over a century. Quantum error correcting codes were recently shown to define a subclass of families of isoclinic subspaces. The Knill–Laflamme theorem is a seminal result in the theory of quantum error correction, a central topic in quantum information. We show there is a generalized version of the Knill–Laflamme result and conditions that applies to all families of isoclinic subspaces. In the case of quantum stabilizer codes, the expanded conditions are shown to capture logical operators. We apply the general conditions to give a new perspective on a classical subclass of isoclinic subspaces defined by the graphs of anti-commuting unitary operators. We show how the result applies to recently studied mutually unbiased quantum measurements (MUMs), and we give a new construction of such measurements motivated by the approach.

In this article, we address the following question: Which hyperbolic or elliptic PDEs admit functional separable solutions. We shall focus on the study of a sinh-Gordon type equation. We construct solutions to this equation via the method of functional separation. We prove that these are the only families that have the property of functional separation and so we obtain a classification. To this end, we construct new families of solutions for the hyperbolic and elliptic versions of both sine and sinh-Gordon equations in a unified way.

A biquandle is a solution to the set-theoretical Yang–Baxter equation, which yields invariants for virtual knots such as the coloring number and the state-sum invariant. A virtual biquandle enriches the structure of a biquandle by incorporating an invertible unary map. This unary operator plays a crucial role in defining the action of virtual crossings on the labels of incoming arcs in a virtual link diagram. This leads to extensions of invariants from biquandles to virtual biquandles, thereby enhancing their strength.

In this article, we establish a connection between the coloring invariant derived from biquandles and virtual biquandles. We prove that the number of colorings of a virtual link L by virtual biquandles can be recovered from colorings by biquandles. We achieve this by proving the equivalence between two different representations of virtual braid groups. Furthermore, we introduce a new set of labeling rules using which one can construct a presentation of the associated fundamental virtual biquandle of L using only the relations coming from the classical crossings. This is an improvement to the traditional method, where writing down a presentation of the associated fundamental virtual biquandle necessitates noting down the relations arising from the classical and virtual crossings.

If ${\mathcal {E}}, {\mathcal {F}}$ are vector bundles of ranks $r-1,r$ on a smooth fourfold X and $\mathop {\mathcal Hom}({\mathcal {E}},{\mathcal {F}})$ is globally generated, it is well known that the general map $\phi : {\mathcal {E}} \to {\mathcal {F}}$ is injective and drops rank along a smooth surface. Chang improved on this with a filtered Bertini theorem. We strengthen these results by proving variants in which (a) ${\mathcal {F}}$ is not a vector bundle and (b) $\mathop {\mathcal Hom}({\mathcal {E}},{\mathcal {F}})$ is not globally generated. As an application, we give examples of even linkage classes of surfaces on $\mathbb P^4$ in which all integral surfaces are smoothable, including the linkage classes associated with the Horrocks–Mumford surface.